ANNALES
UNIVERSIT ATIS MARIAE C U RI E-S K L 0 D O W S K A LUBLIN-POLONIA
VOL. XLVIII, 5________________ SECTIO A__________________________1994
Volker KASTEN (Hannover)
The Sendov-Conjecture and the Maximum Principle
Abstract. If A" is the closed polycylinderthen any x=(zt,...,in)€A"
determinesapolynomialp,(() with leadingcoefficient equal 1 and zerosx*. Let C(i)CA be thesetofcriticalpoints of The author deals withthe properties of the functiond(x)~max{d*(x): t=l,...,n} where<f*(x):= dist(xk ,C(z)). Notethat according to Sendov’s conjecturemax{d(x): x€A"}=1.
1. A reformulation ofthe Sendov-Conjecture
Let A := {z € C : |z| < 1} be the closed unit disk and An := {z= (zj,--- ,2„) £ Cn : |zt| < 1} theclosed unit polycylinder in Cn. The subset
Sn:={(21,--- ,zn)GCn: |zfc| = l, = ,n) C d(A") is called the skeleton of A". For z = (zj,..., z„) G Cn let pz(£) be the normalized polynomial with roots at z*:
Then clearly p2(C) = nn>Ji(C — wj) , where the Wj are the critical points of pz(() . The critical set of p2(£) is defined by C(z) = {wj G C : p2(wj) = 0}. If z G A", then C(z) C A due to the well known Theorem cr GauB-Lucas. In thefollowing we investigate the function d : Cn —* R , defined by
d(z) := max dfc(z) ,
where
d^z) := dist (zt,C(z)) = min {|zt - wj : j = 1, ...,n- 1} . The Conjectureof Sendov can now be formulated as follows:
(*) “ax d(z) = 1 .
x€A"
From a resultof Goodman-Rahman-Ratti [1] on boundary zeros we have
max d(z) = 1 ,
x€Sn
where Sn is the skeleton of A” defined as above. Using this result, (*) can be reformulated as
(**) maxd(z) = max d(z) . ' xgs"
It should be noted that there is a stronger conjecture of Phelps- Rodriguez [2], namely
max d(z) = d(z*) <=> pz*(() = (n — e,a .
x€A"
Up to now this conjecture has been proved for n = 2,3,4 ([2], The
orem 5).
2. A maximum principle for the function d(z)
The formulation (**) of the Sendov-Conjecture involves a max imum principle of some kind for the function d. As is well known, each continuous and plurisubharmonic (p.s.h.) functionon An must attain its maximum value at the skeleton Sn. Unfortunately, logd(z) turns out to be onlypiecewise p.s.h. on A” but fails to be p.s.h. over the whole polycylinder An. This becomes clear from figure 1, where the graph of d(zi, 1, e2"’/3) is plotted over the unit disk |zi| < 1.
Note the boundary maximum with d = 1 at Zi = e4*'/3 , which corresponds to the extremal polynomial p(() = C3 — Phelps- Rodriguez.
54 V. Kasten
Fig.l
Despiteof the fact that d(z) is not plurisubharmonic, it fullfills the following maximum principle, which is essentially due to Phelps andRodriguez ([2], Theorem 2):
Theorem 1. Thefunction d(z) cannot attain a local maximum in Cn .
Sketch of the proof. For z* — (z*,--- ,z*) let C(z*) = {wj,--- ,w*_j} and d(z*) = dist (zj, C(z*)). Then one can move the critical points w' away from z* , holding z* fixed. This can be seen by considering the polynomial
i(C) ■= f
II (3 ~ wi) dz
j=i
If the critical points Wi,... , wn_i of g(£) are chosen near w‘...w£_ j, then theroots zi,...,z„of g(C) arenear zj,...,z*.
Moreover, ifthe critical points w* are moved away from z* , we have d(z) > d(z*).
Theorem 1 implies
max d(z) — max d(z} . z€AB zea(An)
Therefore, inorder toprovethe Sendov-Conjecture (**) it would be sufficient to verify that the function J|a" cannot attain a local maximum at points of 5(An) \ Sn . Formulated in another way, we have the following
Problem 1. Given a polynomial pz*(() such that all its roots ,z* are in A, at least one root is on |^fc| = 1 and at least one root has |^jt| < 1- Can you always find roots zit...,zn near
but within A, such that d(z) > d(z*) ?
3. The restriction of d(z) to lower dimensional planes In order to tackle Problem 1 and also to get a refinement of Theorem 1, it may be helpful to look at the behavior of the function d(z) on lower dimensional planes. It becomes clear from the proof ofTheorem 1, that we have in fact the following
Theorem 1'. Let z* — (z*,...,z*') E Cn and A"-1 be the (n — l^-dimensional plane through z* given by
Then the restriction of d to An~x cannot have a local maximum at z*. As a consequence, d|xn-1 C An attains its maximum on the
“boundary” An-1nd(An).
Remark. In [3],Miller investigatedextremal polynomials p2*(C) for which the restriction of d(z) to {z 6 Cn : Zi = z*} Cl A" is maximal at z*. He proved that 2r + s > n — I, where s denotes the number ofroots of the extremal polynomial on the unit circle and r is the number of critical points on the circle |w — z* | = d(z*) which is called the critical circle.
56 V. Kasten
From Theorem 1' we are led to
Problem 2. Is it true that the restriction d|xn-‘ °f d to an arbitrary (n--- 1)-dimensional plane An_1CCn with A"-1DA"
0 attains its maximum at points of An_1 D9(A”) ? A partialsolution of Problem 2 is given by
Theorem 2. Let A2 C C3 be an arbitrary complex plane of dimension 2 with A2 ClA3 / 0 . Then
max d(z) = max d(z) . z€AJnA3 z€X2na(A3)
Proof. Let z* E A2 Cl A3 such that d(z*) = maxl€j4»nA3 d(z) =: d*.
If 2* 6 A2 Cl 9(A3) , there is nothing to prove. Suppose therefore that z* = (zj, Zj,2J) is an inner point of A3. We may assume that d* = dist (z*,C(z*)) . In the following it will be shown that there exists an at least one-dimensional analytic set S C C3 through z*
with d|s(2) S d* . As is well known, each such set S must intersect d(A3) (cf. [4], Chapt. 2, Corollary 4), and thus the conclusion of Theorem 2follows. In orderto prove theexistence of such an analytic set S we will distinguish two cases, depending on the critical points w*,w? of the extremal polynomial pz*(C) = nt=i (C ~ 2£) •
Case 1: |zj — w*| = |zj — wj|.
Note that the polynomial pz(() = rit=i (C— zk) has the critical points
«>1,2 = 3(^(2) ± yja2(z) -3a2(2) ) , with »
O-l(z) = 2j + Z2 +23, <72(2) = 2i22 + 2123 + Z2Z3 . Consider first the particular case wj / wj. Then er2(z*) —3<72(z*) 0 and thus there is a unique number t such that
to - ¡»ito))* = -y(<4to) - ^to)) •
It followsfrom |-z* — w*| = |z* — wj| that the vectors
±\/<rf(z*') — 3<t2(2*) have tobeorthogonalto the vector z*—j<7i (2*), hence t must be real. With t as above define
M = {z = (zi,z2tz3) € C3 : (21 -jai(z))2 = -^-(^(2) - 3a2(z)) }
and let S be an irreducible component of the analytic set A2 ClM through 2*. Clearly z* 6 S and
dimz* S > dimz* A2 + dimz*M —3 > 1 . By construction,we have
zi - “ 3<t2(2) (*€S)
and therefore
|zi - wi(z)| = |zi -w2(z)| = |(^ - j)^i(2) - 3<7j(z)| (2 € 5).
Since ¿1^2 has a maximum at z* 6 S, the same is true for the restriction d\s- Because y/a^z) — 3a2(z) 0 0 has a holomorphic branchnear 2*, it follows from the maximum principle for holomor
phic functions on analyticsets (cf. [4], Chapt. 4, Theorem 2 G), that a2 (2) — 3<t2(2) is a constant on S and therefore
¿(2) = |2i -Wl(2)| = |2i -w2(2)| = d* (2 G S) .
In the particular case to* =Wj we have — 3a2(z*) = 0 and one can proceed similarly. Define M in this case by
M = {2 G C3 : 0-^(2) — 3a2(z) = 0}
and let 5 again be an irreducible component of A2 (1M through Then clearly 2* G S and
2 .
¿(2) = |2i - W!(2)| = |2, - w2(2)| = |2! - ~<Tl(2)| (2 G 5) .
58 V. Kasten
From this we conclude d(z) = d* on S in view ofthe maximum principle.
Case 2: d* = |z* — wj| < |z* — wj| • In this case we have
d(z) = |zi - ~ ~3<x2(*) I
locally on A2 near the point z*, with an appropriate branch of the root.
Because d\^t has alocal maximum at z*\ the maximum prin
ciple yields
zi - ~ IxAiC-2) “ 3<t2(2) = const
locallynear z* on A2. It follows that |zi —wi(z)| = d* on A2 PlA3. Therefore the conclusion ofTheorem 2follows if d(z) = |zi — wi(z)|
on A2 PlA3. If d(z) |zj — u?i(z)| on A2 nA3, there must be a point z G A2 n A3 such that
|zi -Wi(z)| = |zi - w2(z)| = d* .
But then we are done due to the case 1, which completes the proof of Theorem 2.
Problem 3. Determine the greatest codimension k such that the following is true: If An~k is an arbitrary complex plane ofdimension
n — k with An~k nA" / 0 , then
max d(z) = max d(z) .
i6A"-‘nAn z€An-*na(An) '
With k as in the problem 3, k — n—1wouldimply (**) andthere fore the Sendov-Conjecture. However, in generalk < n — 1. Thiscan beseen from the figure 2, which deals with degreen = 8. Shown there is the graph of d(zx,z2,..., z£) plotted over the unit disk |zi| < 1, for z2 — 1» z3,4 = e±,,r/6, z£6 = e±"r/3, z*8 = ±i. According to thefigure 2, the restriction of d(z) to the one-dimensional z\-plane has maxima at inner pointsbut not on the boundaryofthe unit disk.
l-i
Fig. 2
REFERENCES
[1] Goodman, A.W., Q.I. Rahman andJ.S. Ratti, On the zeros of a polynomial and its derivative,Proc. AMS, 21 (1969), 273-274.
[2] Phelps, D., and R.S.Rodriguez, Some properties of extremal poly
nomials for the Ilieff Conjecture, Kodai Math. Sem. Rpt. 24 (1972), 172-175.
[3] Mil 1er,M.J., Continuous independence and the Meff-Sendov conjec ture, Proc. AMS 115 (1992), 79-83.
[4] Whitney, H., Complex Analytic Varieties, Addison-Wesley Publ. Co., Reading, Massachusetts1972.
Volker Kasten
Mathematisches Institut der Universitt Hannover Welfengarten 1
30167 Hannover, F.R.G.
